In the card game bridge, each of 4 players is dealt a hand of 13 of the 52 cards. What is the probability that each player receives exactly one Ace?
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To find the probability that each player receives exactly one Ace in a game of bridge, we need to calculate the total number of favorable outcomes and divide it by the total number of possible outcomes.
First, let's calculate the total number of favorable outcomes, which is the number of ways each player can receive exactly one Ace.
The first player can receive any of the 4 Aces, so there are 4 choices. Once the first Ace is assigned, there are 48 cards left for the remaining 12 cards in the hand of the first player.
For the second player, there are 3 remaining Aces to choose from, and then 36 cards left for the remaining 12 cards in their hand.
Similarly, for the third player, there are 2 remaining Aces to choose from, and then 24 cards left for the remaining 12 cards in their hand.
Finally, for the fourth player, there is only 1 Ace left, and then 12 cards left for the remaining 12 cards in their hand.
To calculate the total number of favorable outcomes, we multiply the number of choices at each step:
Number of favorable outcomes = 4 * 3 * 2 * 1 * 48 * 36 * 24 * 12 = 29,421,312
Next, let's calculate the total number of possible outcomes, which is the total number of ways to distribute 13 cards to each player.
To do this, we start with 52 cards and distribute them one by one to the four players. The first player can receive any of the 52 cards, the second player can receive any of the remaining 51 cards, and so on.
To calculate the total number of possible outcomes, we multiply the number of choices at each step:
Number of possible outcomes = 52 * 51 * 50 * ... * 41 = 39,029,188,884,700
Finally, we can calculate the probability by dividing the number of favorable outcomes by the number of possible outcomes:
Probability = Number of favorable outcomes / Number of possible outcomes = 29,421,312 / 39,029,188,884,700
Therefore, the probability that each player receives exactly one Ace in a game of bridge is approximately 0.0000755 or 0.00755%.